Theorem ragperp 28810

Description: Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)

Hypotheses

Ref	Expression
isperp.p	⊢ 𝑃 = (Base‘𝐺)
isperp.d	⊢ − = (dist‘𝐺)
isperp.i	⊢ 𝐼 = (Itv‘𝐺)
isperp.l	⊢ 𝐿 = (LineG‘𝐺)
isperp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
ragperp.b	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
ragperp.x	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
ragperp.u	⊢ (𝜑 → 𝑈 ∈ 𝐴)
ragperp.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
ragperp.1	⊢ (𝜑 → 𝑈 ≠ 𝑋)
ragperp.2	⊢ (𝜑 → 𝑉 ≠ 𝑋)
ragperp.r	⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))

Assertion

Ref	Expression
ragperp	⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Proof of Theorem ragperp

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isperp.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		isperp.d	. . . 4 ⊢ − = (dist‘𝐺)
3		isperp.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		isperp.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2740	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		isperp.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ TarskiG)
8		ragperp.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐵 ∈ ran 𝐿)
10		simprr 778	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
11	1, 4, 3, 7, 9, 10	tglnpt 28642	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝑃)
12		isperp.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
13	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ ran 𝐿)
14		ragperp.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
15	14	elin1d 4140	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16	15	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝐴)
17	1, 4, 3, 7, 13, 16	tglnpt 28642	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝑃)
18		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐴)
19	1, 4, 3, 7, 13, 18	tglnpt 28642	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝑃)
20		ragperp.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
21	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝐵)
22	1, 4, 3, 7, 9, 21	tglnpt 28642	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝑃)
23		ragperp.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
24	23	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝐴)
25	1, 4, 3, 7, 13, 24	tglnpt 28642	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝑃)
26		ragperp.r	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
27	26	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
28		ragperp.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≠ 𝑋)
29	28	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ≠ 𝑋)
30	23	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ 𝐴)
31	6	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐺 ∈ TarskiG)
32	17	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝑃)
33	19	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝑃)
34		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → ¬ 𝑋 = 𝑢)
35	34	neqned 2942	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ≠ 𝑢)
36	12	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 ∈ ran 𝐿)
37	15	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝐴)
38	18	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝐴)
39	1, 3, 4, 31, 32, 33, 35, 35, 36, 37, 38	tglinethru 28729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 = (𝑋𝐿𝑢))
40	30, 39	eleqtrd 2842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ (𝑋𝐿𝑢))
41	40	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑢 → 𝑈 ∈ (𝑋𝐿𝑢)))
42	41	orrd 869	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑢 ∨ 𝑈 ∈ (𝑋𝐿𝑢)))
43	42	orcomd 877	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑈 ∈ (𝑋𝐿𝑢) ∨ 𝑋 = 𝑢))
44	1, 2, 3, 4, 5, 7, 25, 17, 22, 19, 27, 29, 43	ragcol 28792	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑉”⟩ ∈ (∟G‘𝐺))
45	1, 2, 3, 4, 5, 7, 19, 17, 22, 44	ragcom 28791	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑉𝑋𝑢”⟩ ∈ (∟G‘𝐺))
46		ragperp.2	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑋)
47	46	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ≠ 𝑋)
48	20	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ 𝐵)
49	6	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐺 ∈ TarskiG)
50	17	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝑃)
51	11	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝑃)
52		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → ¬ 𝑋 = 𝑣)
53	52	neqned 2942	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ≠ 𝑣)
54	8	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 ∈ ran 𝐿)
55	14	elin2d 4141	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
56	55	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝐵)
57	10	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝐵)
58	1, 3, 4, 49, 50, 51, 53, 53, 54, 56, 57	tglinethru 28729	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 = (𝑋𝐿𝑣))
59	48, 58	eleqtrd 2842	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ (𝑋𝐿𝑣))
60	59	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑣 → 𝑉 ∈ (𝑋𝐿𝑣)))
61	60	orrd 869	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑣 ∨ 𝑉 ∈ (𝑋𝐿𝑣)))
62	61	orcomd 877	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑉 ∈ (𝑋𝐿𝑣) ∨ 𝑋 = 𝑣))
63	1, 2, 3, 4, 5, 7, 22, 17, 19, 11, 45, 47, 62	ragcol 28792	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑣𝑋𝑢”⟩ ∈ (∟G‘𝐺))
64	1, 2, 3, 4, 5, 7, 11, 17, 19, 63	ragcom 28791	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
65	64	ralrimivva 3183	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
66	1, 2, 3, 4, 6, 12, 8, 14	isperp2 28808	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))
67	65, 66	mpbird 258	1 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054   ∩ cin 3889   class class class wbr 5079  ran crn 5626  ‘cfv 6492  (class class class)co 7363  ⟨“cs3 14802  Basecbs 17177  distcds 17227  TarskiGcstrkg 28520  Itvcitv 28526  LineGclng 28527  pInvGcmir 28745  ∟Gcrag 28786  ⟂Gcperpg 28788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-er 8640  df-map 8772  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607  df-hash 14291  df-word 14474  df-concat 14531  df-s1 14557  df-s2 14808  df-s3 14809  df-trkgc 28541  df-trkgb 28542  df-trkgcb 28543  df-trkg 28546  df-cgrg 28604  df-mir 28746  df-rag 28787  df-perpg 28789

This theorem is referenced by:  footexALT  28811  footexlem2  28813  colperpexlem3  28825  mideulem2  28827  lmimid  28887  hypcgrlem1  28892  hypcgrlem2  28893  trgcopyeulem  28898